Jeremy Corbyn has accused Theresa May of seeking to "segregate" the country with her plans to create a new wave of grammar schools.

The Labour leader was cheered by his backbenchers at Prime Minister's Questions on Wednesday as he warned the move would lead to "second-class schooling for the many".

But May hit back, accusing Corbyn of trying to "pull up the ladder" behind him after enjoying his own grammar school education as a boy.

The prime minister, who also went to a grammar school, robustly defended her plans to allow state schools to select pupils on the basis of academic ability.

She has come under fire from teaching unions, the outgoing head of schools watchdog Ofsted, and some Conservative MPs over the proposals amid warnings that poorer pupils will lose out.

Corbyn used all of his allotted six questions to press May on the issue at PMQs. He said: "Isn't this the case of a government heading backwards to a failed segregation for the few and second-class schooling for the many?

Labour MPs laughed as Corbyn congratulated May on the "new era of unity" she had unwittingly created among opponents to her grammar school plans. He asked her to name any educational experts who actually backed her.

And he concluded: "Isn't all this proof that the Conservative party's green paper addresses none of the actual crises facing our schools system – a real-terms cut in school budgets, half a million pupils in super-size classes, a crisis in teacher recruitment and retention, a rising number of unqualified teachers in classrooms, vital teaching assistants losing their jobs?"

But May insisted the plans would improve life chances for 1.25 million children in "not good or outstanding" schools. Reminding Corbyn that he himself had gone to a grammar school, she accused Labour MPs of "taking the advantage of a good education for themselves and pulling up the ladder behind them".

May said: "He went to a grammar school, I went to a grammar school. It is what got us to where we are today – but my side might be rather happier about that than yours."

Corbyn hit back: "It’s not about pulling up ladders, it’s about providing a ladder for every child."